Exponential Models - Constant Growth Rate Model

Activity #7
Exponential Models
Constant Growth Rate Model

When looking at population growth over a short period of time it may appear to follow a linear function.

Optional: Use the U.S. Population Clock and record the U.S. population at regular intervals such as every 1/2 hour for a few hours or every 24 hours for a week. Determine the best function for this data.

While a linear function can be used to model population growth that has a constant increase or decrease in the number of people, an exponential function can be used to model population growth that has a constant percentage change in population.

Since the average annual percent change in a population (growth rate) is often relatively constant during a short period of time, it is not uncommon to fit an exponential model to population data.

Exponential Function
Constant Growth Rate Model

f(t) = ab^t

^f(t)^{= population after t years

a = initial value

b = base or growth factor

t = time in years}

This exponential model can be used to predict population during a period when the population growth rate remains constant.

From the U.S. Census Bureau's Historical National Population Estimates, 1900 to 1999, record the estimated national population in 1999 and the estimated average annual percent change (growth rate given in percent) for that year.

Assuming that the growth rate remains constant, give an equation to express the population as a function of t, the number of years after 1999. Remember:

f(t) = Population t years after 1999

a = Population in 1999

b = Growth factor = 1 + growth rate in decimal form

t = number of years after 1999

Example: In July 1964 the population was 191,888,791 and the growth rate was +1.39 % or 0.0139.

The growth factor = 1 + 0.0139 = 1.0139

This means that the population in 1964 was approximately 1.0139 times the population in 1963.

An equation to find the population t years after 1964 would be:

f(t) = (191,888,791)(1.0139)^t

Using the equation you determined for 1999 data, predict the population in 2000, the current year, and 2010.
Compare your results to the estimated values given in the International Database (IDB) Summary Demographic Data for the U.S. as well as the U.S. Population Clock. How close were your results? Why might they be different?
Using the U.S. census data Population, Housing Units, Area Measurements, and Density: 1790-1990 that you previously assembled and the average annual growth rates that you calculated, record the national population for any year prior to 1970 and the average growth rate for that year. Assuming that the growth rate remained at this value, give an equation to express the population as a function of t, the number of years after your selected year.
Using this equation, predict the population in 2000, the current year, and 2010.
Compare your results to the estimated values given in the International Database (IDB) Summary Demographic Data for the U.S. as well as the U.S. Population Clock. How close were your results? Why might they be different?
From what you've learned, is this exponential model good for predicting population in the short term? In the long term? What about over thousands of years? Explain your reasoning.
Optional: Using the constant growth rate model and U.S. population and growth rate data from 1999, predict when the U.S. population will reach 300 million. Assume the growth rate remains constant. When will it reach 350 million? When will the population double in size from its present value? Why might your results differ from those given in the International Database (IDB) Summary Demographic Data ?

Activity #7 Exponential Models Constant Growth Rate Model

Activity #7
Exponential Models
Constant Growth Rate Model